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Semigroup Methods in Ring Theory* View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 12, 177-190 (1969) Semigroup Methods in Ring Theory* FRANK ECKSTEIN Mathematisches Institut der Universitrit n/3iinster, 44 Miinster, West Germany Communicated by A. W. Goldie Received July, 1968 Let A be a ring and denote by R(A) the Jacobson radical. We will write R instead of R(A) if it is clear from the context in which ring we operate. The question of lifting idempotents of the ring A/R to the ring A arises in various contexts in ring theory. However, what one would frequently like to know beyond the existence of idempotents in A mapping onto a given set of idempotents in A/R is whether or not a given set of orthogonal (summable) idempotents of A/R can be lifted orthogonally (summably) to d. In general these questions have a negative answer. There arc rings which contain idempotents modulo the radical which cannot be lifted. There are rings containing orthogonal idempotents modulo the radical which cannot be lifted orthogonally, inspite of the fact that each single idempotent can be lifted. For an example see ([13], 3.A). It will be shown, however, that any countable set of orthogonal idempotents of A/R can be lifted orthogonally to A provided the individual idempotents can be lifted. The approach to the indicated problem will be semigroup theoretical. Extensive use will be made of the structure theory of completely simple semigroups ([5], pp. 76). This procedure has several advantages. Firstly, it leads to more general results, even in very simple cases [compare e.g. ([7], Prop. 5, p. 54) with (Corollary 18). K o reference to the actual lifting is needed]. Secondly, the results are obtained in a more conceptual way. Almost all of the often very tricky calculations can be avoided (compare e.g. [8], Lemma 12, p. 166 and (22) or [13], 4.6 and (21)). Unfortunately our method will not be very helpful in the lifting of single idempotents. It turns out that the lifting of an idempotent is tantamount to the existence of a completely simple minimal ideal in a certain semigroup. In general, however, this is an equally inaccessible problem. On the contrary * This paper is part of the auther’s dissertation which was written at Tulane University under the supervision of Professor K. H. Hofmann. 177 178 ECKSTEIN for a special class of semigroups the described correspondence will allow us to conclude that a minimal ideal exists. H. Leptin showed in ([9], Th. 19, p. 260) that any linearly compact ring admits a hyperdirect decomposition into two-sided massive ideals. For rings with identity we can get the same result under a somewhat weaker condition than linear compactness. In ([IO], Th. 2, p. 42) A. Malcev proved that in a finite dimensional algebra which is separable modulo its radical any two complements of the radical are conjugate under an inner automorphism. As another application of the method developed in this paper we will shou- in [6] that this theorem can be considerably generalized. I. LIFTING 0F Irmr~onm~rS MODULO THE I*.~DICAL Let S bc a semigroup. A non-empty subset I of S is a left, (u$zt, fwosided) ideal if SIC I (IS C I, SI u IS C I). An ideal K or K(S) is said to be the minimal ideal or the kernel of S if K contains no other ideal properly. A semigroup S is simple if it contains no proper ideals. Let E(S) be the set of idempotents in S, then we introduce a partial order in E(S) by defining e K f if and only if ef = fe = e. An idempotent e is primitive if it is minimal w.r.t. the partial order just defined, i.e. if it is the only idempotent in eSe. A simple semigroup which contains a primitive idempotent is called completely simple. Two elements a, h E S are said to be LP-eqz&aZent, a-P%, respectively .%?-equivalent, a%, if they generate the same left, respectively right ideal. The join of the equivalence relations 2 and .B’ is denoted by ‘9, their intersection by Y. For the algebraic theory of semigroups \ve will refer to the book of Clifford and Preston [5]. All results on semi-groups used in the following can be found there. I 7'IIIWREM. Let A be a ring, B an ideal contained in the vadicul of A and suppose 0 f e .:- N -+ B is an idempotent in A/B. Then (a) The residue &ass S =- x + B is a semigroup w.r.t. the multiplication in A as well as w.r.t. the ~-multiplication, .r~y=r$-y~~yanndife~s then e2 eiffe-e =e. (b) For each idempotent e E S, eSe = e + eBe is a group under ovdi?lary multiplication. (c) Each idempotent e E S is contained in the minimal ideal of S which is completely simple. SEMIGROUP METHODS IN RING THEORY 179 (d) Zf e is an idempotent and N = eBe, then e + N” is a normal subgroup ofe$N. (e) The nth term F,(e + N) of the lowev central series is contained in e -+- Nn. Zt1particular, if B is a nilpotent ideal in A the group e + N is nilpotent. Proof. (a) S is a semigroup w.r.t. the ring multiplication is clear. Let s, f E S then s = x -1. b, t = N $- b’ and s o t = .Y + (x - A?) + b + b’ f bx -t xb -+ bb’ t .I’ + B. (1~) Let N E eBe, then s is quasi-regular. Let y be the quasi-inverse of x, theny E eBe and (e - y) is the inverse of (e - ,x) w.r.t. e, i.e., (e - x)(e -y) = e 7. (e - y)(e - x). Thus each element of e -I- eBe has a twosided inverse which means e -I-. eBe is a group. ‘1’0 prove (c) we distinguish two cases: (I) S = x -/- B contains a zero e. Then x + B contains a minimal ideal, namely e. We have to show e is the only idempotent contained in S = e -1 B. As e is a zero we have e(e +- b) = (e -t- b) e == e. Hence eb = be == 0 for all DEB. Letf- e f b be an idempotent, then e + b = f = f2 = e + b* and h2 = b which means b == 0 as B is contained in the radical of A. Thusf = e. (2) 5’ .:.. m + B does not contain a zero. Then we apply [7] Ex. 14, p. 84 and use (b). (d) First we show e + NT/ is a subgroup. For this we observe that e - u is the inverse of e - Y iff u .‘. r -- ~7%:- 0. Hence if e -- Y E e + NT?, then c: =: rw - Y t -V?l and e - z’ E e + N1. To show that e + N” is normal in c ! iV let e -{ r E e + lVi’ and P $- s, e -+- f E e + IV such that (e s)(e + t) =- P -I- s 2 t + st = e, then (e + s)(e i r)(e $ t) = e + s -i- r + SY i- t t sf + rt + svf = e A r -t sr + rt + srt E e + N”. (e) The group r,,(e $ N) are defined as follows: I’,(e + N) = e -+ N and r,,,de + N) - (r,,(e + W, e + 1V). We proceed by induction: The assertion is clear for n = 1. Eow suppose that e + r, e + r E &(e + V) C e -/- A’“, and E T s, e + s E e -$ N, where (e -1 r)(e j- “) = e + Y f~ r -t YY 1.. e and (c $ s)(e -I- S) :: e + s + s + ss y c, then (e $ r)(e --t s)(e + i)(e + S) = (e I- T + s + rs)(e $- i -$ S + 73) = e + f.? + I-S + rFi + SF + sfi f YS + )‘p -1, US f VSG E e + N”+‘. This finishes the proof. 2 COROLLARY. Let A be a complete, topological ring with ideal neighborhoods of zero, B an ideal contained in the radical of A. Let M be a summable set of 180 ECKSTEIN idempotents in A/B and suppose there exists at least one idempotent e E A mapping onto z E i3,iB for each F E &?. Then any collection M of idempotents in A-at most one out of each residue class t?E k-is summable. Proof. For the definition of a summable family in a topological group vve refer to Bourbaki ([I], pp. 59). It suffices to show that given any ideal neighborhood If of zero, then k’ contains all but finitely many e E M. By hypothesis I- contains almost all E t a module B. Hence for almost all e E M, V A (e 1m B) + ,: Again by hypothesis I,- is a twosided ideal in =1. Thus r n (e -.- B) is a twosided ideal in e + B. Hence the minimal ideal of e j- B is contained in I’ for all but finitely many of the semigroups e + B with e EM. By Theorem 1 this finishes the proof. 3 COROLLARY. Let A be a complete, topological ring with left ideal neighborhoods of zero, B an ideal contained in the radical, let &? be a summable set of idempotents in A/B and suppose there is at least one idempotent e E A mapping onto F E A/B f OYeach .?E M.
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